A variational approach to second order mean field games with density constraints: The stationary case

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain Omega subset of R-d. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order q' is an element of ]1, d/(d - 1) [, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou and Brenier (see [1]), we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order q' >= d (d - 1), the previous arguments do not apply and we prove the existence by means of an approximation argument. (C) 2015 Elsevier Masson SAS. All rights reserved.